3.2.2 Barrier lowering due to image charges


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3.2.2 Barrier lowering due to image charges

Image charges build up in the metal electrode as carriers approach the metal-semiconductor interface. The potential associated with these charges reduces the effective barrier height. This barrier reduction tends to be rather small compared to the barrier height itself. Nevertheless this barrier reduction is of interest since it depends on the applied voltage and leads to a voltage dependence of the reverse bias current. Note that this barrier lowering is only experienced by a carrier which approaches the interface and will therefore not be noticeable in a capacitance-voltage measurement.

An energy band diagram of an n-type silicon Schottky barrier including the barrier lowering is shown in the figure below:

The calculation of the barrier reduction assumes that the charge of an electron close to the metal-semiconductor interface attracts an opposite surface charge which exactly balances the electron's charge so that the electric field surrounding the electron does not penetrate beyond this surface charge. The time to build-up the surface charge and the time to polarize the semiconductor around the moving electron is assumed to be much shorter than the transit time of the electron 1. This scenario is based on the assumption that there are no mobile or fixed charges around the electron as it approaches the metal-semiconductor interface. The electron and the induced surface charges are shown in the figure below:

It can be shown that the electric field in the semiconductor is identical to that of the carrier itself and another carrier with opposite charge at equals distance but on the opposite side of the interface. This charge is called the image charge. The difference between the actual surface charges and the image charge is that the fields in the metal are distinctly different. The image charge concepts is justified on the basis that the electric field lines are perpendicular to the surface a perfect conductor, so that, in the case of a flat interface, the mirror image of the field lines provides continuous field lines across the interface. The electrostatic force between the two particles, one with a positive electronic charge and the other with a negative electronic charge, which are both a distance x away from the interface at x = 0, is given by:

The corrresponding potential equals:

which combined with the potential variation due to the electric field yields the following potential energy, V(x), versus position, x:

where the field due to the charge in the depletion region is assumed to be constant and set equal to the maximum field:

At this point the question arises why the potential can be noticeably altered by a single electron, while the depletion layer contains significantly more charge. To understand this, one has to realize that we have assumed that the charge in the depletion region is not quantized, but instead is distributed throughout the depletion layer. While this assumption does provide the correct average potential it does not accurately reflect the potential variations due to the individual charges of the ionized donors or mobile electrons. As we assumed that the single electron is far away from all other charges in the semiconductor, the potential energy due to all those charges is close to the average potential energy.

The potential energy due to the distributed charge of the ionized donors and a single electron reaches its maximum value at:

and the corresponding maximum value of the potential energy equals:

where DfB is the barrier height reduction given by:


  1. The time to polarize the semiconductor equals to first order the dielectric relaxation time, t = r es, which is typically of the order of 100 fs. A measurement of the barrier lowering as a function of the electric field has been obtained from photoelectric measurements on Au-silicon Schottky barriers and yielded a relative dielectric constant of es/e0 = 12 +/- 0.5.

    S. M. Sze, C. R. Crowell, and D. Kahng, "Photoelectric Determination of the Image Force Dielectric Constant for Hot electrons in Schottky Barriers," J. Appl. Phys., 35, 2534 (1964)

3.2.1 ® 3.2.3

© Bart Van Zeghbroeck 1997